The (real) Lie groups \(SL(2,\mathbb {H})\)  and \(SL(2,\mathbb {O})\) , their respective spinor representations on \(\mathbb {H}^2\)  and \(\mathbb {O}^2\)  and their respective vector representations on \(\mathfrak {h}_2(\mathbb {H})\)  and \(\mathfrak {h}_2(\mathbb {O})\)  can then be defined by exponentiating these Lie algebras (see [1]}, [2]}, [3]}, [4]}). In the following we use the notation (REF ) to represent the action of \(SL(2,\mathbb {K})\)  on \(\mathfrak {h}_2(\mathbb {K})\)  (even in the cases where \(gmg^\dagger \)  does make sense as a matrix product).
